Question

Cho \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{a}\). Chứng minh rằng a=b=c

Answer 1

Ta có:a/b=b/c=c/a (1)

=>(a².c)/abc=(b².a)/abc=(c².b)/abc(quy đồng)

=>a²c=b²a=c².b

Ta lại có: a²c=b²a=>a²/a=b/c=>a=b/c (2)

b²a=c².b=>b²/b=c/a=>b=c/a (3)

a²c=c².b=>c²/c=a/b (4)

Từ (1), (2), (3),(4) => a=b=c (.)

Answer 2

Ta có: \(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{c}{a} \)<1>

=> \(\dfrac{(a^2.c)}{abc}\)=\(\dfrac{(b^2.a)}{abc} \)=\(\dfrac{(c^2.b)}{abc}\)

=> a².c = b².a = c².b

Và a².c = b².a => \(\dfrac{a^2}{a} \)=\(\dfrac{b}{c} \)=> a= \(\dfrac{b}{c}\)<2>

b².a = c².b => \(\dfrac{b^2}{b}=\dfrac{c}{a}\)=> b= \(\dfrac{c}{a}\)<3>

a².c = c².b => \(\dfrac{c^2}{c}=\dfrac{a}{b}\)<4>

Từ <1><2><3><4> => a=b=c